Pressure estimates and classifications for steady H1loc-solutions to two-dimensional incompressible Euler equations
Abstract
We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the L∞∩ H1loc class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that ∇ P∈ Lq for some 1≤ q≤ 2. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to H1loc. The pressure serves as a unifying quantity throughout the analysis.
Related
- Nonuniqueness of Solutions to 2D Incompressible Euler Equations with Unbounded Vorticity
- Stability Analysis of Bounded Solutions to 2-D Incompressible Navier-Stokes Equations in L^2_uloc Spaces
- Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction
- Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces
- Linear Corrector Estimates and Corrected Pressure Convergence for a Two-Parameter Hyperbolic Relaxation of the Incompressible Navier--Stokes Equations
BibTeX
@misc{gui2026pressure,
title = {{Pressure estimates and classifications for steady $H^1_\rm loc$-solutions to two-dimensional incompressible Euler equations}},
author = {Changfeng Gui and Qinfeng Li and Chunjing Xie and Huan Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32647},
eprint = {2609.32647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32647},
}
RIS
TY - UNPB
TI - Pressure estimates and classifications for steady H^1_loc-solutions to two-dimensional incompressible Euler equations
AU - Changfeng Gui
AU - Qinfeng Li
AU - Chunjing Xie
AU - Huan Xu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32647
UR - https://arxiv.org/abs/2609.32647
AB - We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the L^∞∩ H^1_loc class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that ∇ P∈ L^q for some 1≤ q≤ 2. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to H^1_loc. The pressure serves as a unifying quantity throughout the analysis.
ER -